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Homotopical patch theory

Carlo Angiuli, Edward Morehouse, Daniel R. Licata, Robert Harper
2014 Proceedings of the 19th ACM SIGPLAN international conference on Functional programming - ICFP '14  
A patch theory is presented by a higher inductive type. Models of a patch theory are functions from that type, which, because function are functors, automatically preserve the structure of patches.  ...  We reformulate patch theory using the tools of homotopy type theory, and clearly separate formal theories of patches from their interpretation in terms of basic revision control mechanisms.  ...  Building on this work, we develop patch theory in the context of homotopy type theory, using paths to model aspects of patch theory.  ... 
doi:10.1145/2628136.2628158 dblp:conf/icfp/AngiuliMLH14 fatcat:oqd3onn2jvh3nnbipg4hfi7dne

Planar Graphs on Nonplanar Surfaces

Bojan Mohar, Neil Robertson
1996 Journal of combinatorial theory. Series B (Print)  
Suppose now that at each vertex of C, patch faces are only on one side of C. Then there is a cycle C homotopic to C that contains only vertices in the interiors of patches.  ...  Since # cannot escape out of D, it is homotopic to # 1 and it must intersect # 1 (or # 2 ) twice. Let x$ be the other patch vertex used by #.  ... 
doi:10.1006/jctb.1996.0058 fatcat:lvlzteyqbfcbxloihhxzisisgi

Page 305 of Mathematical Reviews Vol. 57, Issue 1 [page]

1979 Mathematical Reviews  
He then formulates ‘‘a simple rule for identifying theories that admit physically significant patching ambiguities”.  ...  Preprint, Harvard Univ., Cam- bridge, Mass., 1975] has given rules for patching together two topologically stable field configurations in non-Abelian gauge theories (that is, constructing models of a two-particle  ... 

More (thoughts on) Gribov copies

Pierre van Baal
1992 Nuclear Physics B  
In ref. [6] we proposed formulating the hamiltonian theory on coordinate patches with homotopically non-trivial gauge transformations as transition functions.  ...  We can shrink these patches almost to A (and their associated gauge copies, with the homotopically non-trivial gauge transformations that relate the inequivalent classical vacua).  ... 
doi:10.1016/0550-3213(92)90386-p fatcat:wfdt6tdp3fgltgnziihzqsavuy

Tōhoku

Rick Jardine
2015 Inference: International Review of Science  
W e now understand the theory of stacks and non-abelian cohomology theory in a much more tractable, homotopical way. Stack theory is, I suggest, the modern geometric theory of symmetries.  ...  We end with a purely homotopical approach to the construction of the derived category of abelian sheaves, one that is consistent with the homotopical theory of simplicial sheaves and presheaves in the  ... 
doi:10.37282/991819.15.13 fatcat:3pmvwf64pzc5vddgj5t7zrstfu

Homotopy Transition Cocycles [article]

James Wirth, Jim Stasheff
2006 arXiv   pre-print
The full theory was worked out by the first author in his 1965 Notre Dame thesis wirth:diss. Here we present it using language that has been developed in the interim.  ...  trivial fibrations, one can define transition functions g : U∩ U→ H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that g g is homotopic  ...  In Section 5, Wirth's concept of a "fibration theory" is axiomatized. Here too the patching theorem is crucial.  ... 
arXiv:math/0609220v2 fatcat:p7zdry3nenaw5h3uhvhcsxsboq

An Approximation Theorem for Maps Between Tiling Spaces [article]

Betseygail Rand, Lorenzo Sadun
2018 arXiv   pre-print
If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.  ...  Introduction Many aspects of tiling theory, such as pattern-equivariant cohomology [2, 3] , are built around local data.  ...  central patches of radius 1.  ... 
arXiv:0906.4741v2 fatcat:fvuoiduydrfulgpdaz7t5hdurm

Special issue dedicated to ICFP 2014: Editorial

DEREK DREYER, MARY SHEERAN
2016 Journal of functional programming  
demonstrate both the quality and the breadth of the conference, with a strong emphasis on types and their applications, and ranging from compilation methods through contract verification to homotopy type theory  ...  In Homotopical Patch Theory, Angiuli, Morehouse, Licata, and Harper consider a programming application of higher inductive types, a new class of datatypes that arises in homotopy type theory.  ...  The paper considers a sequence of patch theories, including two that did not appear in the original conference version of the paper, culminating in a patch theory of text files.  ... 
doi:10.1017/s0956796816000228 fatcat:raf3t4hb3jahthr3ngcb5oo6zq

Topological Membrane Solitons and Loop Orders of Membrane Scatterings in M-theory [article]

Chien-Hao Liu
1996 arXiv   pre-print
Patching of these solitons and their topological charges are also defined and discussed. (2) Loop order of membrane scatterings is the basis for a perturbative M-theory.  ...  Two topological issues on membranes in M-theory are studied: (1) Soliton is an important subject in M-theory.  ...  The additivity property of charges under patchings still holds. As a comparison, the patching of membrane solitons defined here generalizes the patching that occurs in gauge theory.  ... 
arXiv:hep-th/9610042v1 fatcat:dxgngnoh3jhatithhxemfbe5gm

Page 548 of The American Mathematical Monthly Vol. 58, Issue 8 [page]

1951 The American Mathematical Monthly  
Whereas a cycle is homologous to zero if it bounds, it is homotopic to zero if it can be contracted into a point by continu- ous deformation.  ...  So much, however, seems clear that one had better start, not with a division into cells, but with a covering by patches which are allowed to overlap.  ... 

Global Color Is Not Always Defined

Philip Nelson, Aneesh Manohar
1983 Physical Review Letters  
Far away from a fundamental magnetic monopole of the SU(5) grand unified theory, the full theory reduces to one whose gauge group is unbroken SU(3) ⊗ U(1).  ...  Disciplines Physical Sciences and Mathematics | Physics Comments ABSTRACT Far away from a fundamental magnetic monopole of the SU(5) grand unified theory, the full theory reduces to one whose gauge group  ...  To proceed beyond this heuristic level it is convenient to 4 describe the unbroken theory in Wu and Yang's two-patch formalism.The theory is defined on the space 1R x OR 3 -B }, where B is a ball occupied  ... 
doi:10.1103/physrevlett.50.943 fatcat:aohk5swrkzhkxm5zr7h7bw7nwi

Page 8930 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Some important flows such as vortex patches belong to this class.” J.  ...  Near the absolute instability threshold, spatial growth rates are larger than those predicted by temporal stability theory.  ... 

Page 4428 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
Theory Appl. 114 (2002), no. 3, 609-637.  ...  This attempt opens a route which, on further examination, might eventually reach unexplored patches in the field.  ... 

Global signatures of gauge invariance: Vortices and monopoles

H. C. Tze, Z. F. Ezawa
1976 Physical Review D, Particles and fields  
By way of homotopy theory, a simple analysis is presented for the global groups U (1)) O(3), SU(2) then SU(N)/ZN and SU(N).  ...  A comprehensive topological classification of vortices and their endpoint Dirac monopoles is formulated in gauge theories with an arbitrary compact Lie group.  ...  One of us (HCT) wishes to thank his colleagues at the SLAC Theory Group for many discussions.  ... 
doi:10.1103/physrevd.14.1006 fatcat:7gheplrs6ngktfdct4qzics4za

Desingularizing homology manifolds

J L Bryant, Steven Ferry, Washington Mio, Shmuel Weinberger
2007 Geometry and Topology  
In earlier work, [4] , we showed that nonresolvable homology manifolds exist and can be classified, up to s-cobordism, by a variant of surgery theory.  ...  We may assume that f restricts to U V 1 maps on the patches of X .  ... 
doi:10.2140/gt.2007.11.1289 fatcat:g7zbf3btgzcu5jguljwzh7yrra
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